# Electric charges and magnetic monopoles in Gravity's RainbowAcademic research paper on "Physical sciences"

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## Abstract of research paper on Physical sciences, author of scientific article — Remo Garattini, Barun Majumder

Abstract In this work, we explore the possibility that quantum fluctuations induce an electric or magnetic charge or both, in the context of Gravity's Rainbow. A semi-classical approach is adopted, where the graviton one-loop contribution to a classical energy in a background spacetime is computed through a variational approach with Gaussian trial wave functionals. The energy density of the graviton one-loop contribution, in this context, acts as a source for the electric/magnetic charge. The ultraviolet (UV) divergences, which arise analyzing this procedure, are kept under control with the help of an appropriate choice of the Rainbow's functions. In this way we avoid the introduction of any regularization/renormalization scheme. A comparison with the observed data leads us to determine the size of the electron and of the magnetic monopole which appear to be of Planckian size. Both results seem to be of the same order for a Schwarzschild and a de Sitter background, respectively. Estimates on the magnetic monopole size have been done with the help of the Dirac quantization procedure. We find that the monopole radius is larger than the electron radius. Even in this case the ratio between the electric and magnetic monopole radius appears to be of the same order for both geometries.

## Academic research paper on topic "Electric charges and magnetic monopoles in Gravity's Rainbow"

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Nuclear Physics B 883 (2014) 598-614

www. elsevier. com/locate/nuclphysb

Electric charges and magnetic monopoles in Gravity's Rainbow

Remo Garattiniab, Barun Majumderc*

a Universita degli Studi di Bergamo, Dipartimento di Ingegneria, Viale Marconi 5, 24044 Dalmine (Bergamo), Italy

b I.N.F.N. - Sezione di Milano, Milan, Italy c Indian Institute of Technology Gandhinagar, Ahmedabad, Gujarat 382424, India

Received 24 February 2014; accepted 8 April 2014

Available online 13 April 2014

Editor: Stephan Stieberger

Abstract

In this work, we explore the possibility that quantum fluctuations induce an electric or magnetic charge or both, in the context of Gravity's Rainbow. A semi-classical approach is adopted, where the graviton one-loop contribution to a classical energy in a background spacetime is computed through a variational approach with Gaussian trial wave functionals. The energy density of the graviton one-loop contribution, in this context, acts as a source for the electric/magnetic charge. The ultraviolet (UV) divergences, which arise analyzing this procedure, are kept under control with the help of an appropriate choice of the Rainbow's functions. In this way we avoid the introduction of any regularization/renormalization scheme. A comparison with the observed data leads us to determine the size of the electron and of the magnetic monopole which appear to be of Planckian size. Both results seem to be of the same order for a Schwarzschild and a de Sitter background, respectively. Estimates on the magnetic monopole size have been done with the help of the Dirac quantization procedure. We find that the monopole radius is larger than the electron radius. Even in this case the ratio between the electric and magnetic monopole radius appears to be of the same order for both geometries.

* Corresponding author.

E-mail addresses: Remo.Garattini@unibg.it (R. Garattini), barunbasanta@iitgn.ac.in (B. Majumder). http://dx.doi.Org/10.1016/j.nuclphysb.2014.04.005

1. Introduction

It was Andrei Sakharov in 1967 [1] who first conjectured the idea of Induced gravity (or emergent gravity), namely spacetime background emerges as a mean field approximation of underlying microscopic degrees of freedom, similarly to the fluid mechanics approximation of Bose-Einstein condensates. This means that some basic ingredients of General Relativity like the gravitational Newton's constant can be computed by means of quantum fluctuations of some matter fields. This idea is opposed to the concept of "charge without charge" and "mass without mass" arising from the spacetime foam picture of John A. Wheeler [2], where the matter properties emerge as a geometrical feature of spacetime. In a foamy spacetime topological fluctuation appear at the Planck scale, meaning that spacetime itself undergoes a deep and rapid transformation in its structure. Wheeler also considered wormhole-type solutions as objects of the quantum spacetime foam connecting different regions of spacetime at the Planck scale. Although the Sakharov approach has the appealing property of being renormalizable "ab initio" because it involves only quantum fluctuations of matter fields described by bosons and fermions, the Wheeler picture involves quantum fluctuations of the gravitational field alone and since one of the purposes of Quantum Gravity should be a realization of a theory combining Quantum Field Theory with General Relativity, it appears that spacetime foam is the right candidate for such a description. Unfortunately, every proposal of Quantum Gravity except string theory has to face with Ultra Violet (UV) divergences. Recently a proposal which uses a distortion of the gravitational field at the Planck scale, named as Gravity's Rainbow [3,4] has been considered to compute Zero Point Energy (ZPE) to one loop [5,6]. The interesting point is that such a distortion enters into the background metric and becomes active at Planck's scale keeping under control UV divergences. Briefly, the situation is the following: one introduces two arbitrary functions gi (E/Ep) and g2 (E/Ep), denoted as Rainbow's functions, with the only assumption that

lim g1(E/EP) = 1 and lim g2(E/EP) = 1. (1)

E/EP E/EP

On a general spherical symmetric metric such functions come into play in the following manner

o o dt dr r / ry ry /->,

ds2 = -N 2(r) —-+ —-T-— + —-(dd2 + sin2 0d02), (2)

()g2(E/EP) g2(E/EP)(1 - g\(E/EP)K Vh "

where N(r) is the lapse function, b(r) is denoted as the shape function and EP is the Planck energy. The purpose of this paper is to approach one of the aspects of Wheeler's ideas, namely "charge without charge". In particular, we will investigate if quantum fluctuations of the gravitational field can be considered as a source for the electric/magnetic charge. Note that a similar approach to realize "charge without charge" has been described in Ref. [7]. However due to UV divergences a regularization/renormalization was used to obtain finite results and if a renormal-ization group like equation has been used, the final result would depend on the renormalization point scale Here the renormalization point is fixed at the Planck scale due the Rainbow's functions. It is clear that, if an electric/magnetic charge can be generated, this information is encoded in the Einstein's field equations. These equations are simply summarized by

R/J.V - ^g^vR = (3)

_ 1 T^V = 4n

F^yFY 4 g^vFygFr

is the energy-momentum tensor of the electromagnetic field, F,iV = B^AV - dvAk = 8nG, with G the Newton's constant and here we have neglected the contribution of the cosmological constant Ac. The electromagnetic field strength tensor F^v can be computed with the help of the electromagnetic potential A^ which, in the case of a pure electric field assumes the form A^ = (Qe/r, 0, 0, 0) while in the case of pure magnetic field, the form is A^ = (0, 0, 0, -Qm cos 0). Qe and Qm are the electric and magnetic charge respectively. It is interesting to note that Qe and Qm contribute in the same way to the electromagnetic Hamiltonian density. Indeed, for the electric charge, the on-shell contribution of Tapuau@ is

,, v 1 2 1 Q2 T^uW = — (F01)2 = — Q = Pe, (5)

8n 8n r4

and when we consider the magnetic charge, we get

u»uv = — (F23)2 = — Qm = Pm- (6)

1 (F )2 1 Q2m

u _ —(F23) = --T

uM is a time-like unit vector such that u ■ u = —1. However, while the electric charge exists, for the magnetic charge or magnetic monopole, there is no experimental evidence for its existence.1 The magnetic monopole search has a long history in theoretical physics: predicted by Paul Dirac in 1931, he showed that QED allows the existence of point-like magnetic monopole with charge

Qm = — (7)

or an integer multiple of it [10]. Subsequently this prediction was also confirmed by Gerard 't Hooft and Alexander Polyakov who showed that magnetic monopoles are predicted by all Grand Unified Theories (GUTs) [11]. Although monopoles of grand unified theories would have masses typically of the order of the unification scale (m ~ 1016 GeV) but generally there are no tight theoretical constraints on the mass of a monopole. For this reason, the reference value of our calculation will be that of the electric charge. It is important to remark that in a system of units in which h = c = k = 1, that will be used throughout the paper2

e2 =---(9)

137 V ;

The rest of the paper is organized in the following manner. In Section 2 we introduce the charge operator, in Section 3 we introduce the charge operator in presence of Gravity's Rainbow specified to the Schwarzschild and to the de Sitter metric, in Section 4 we will apply the charge operator to the magnetic monopole case and in Section 5 we will summarize and conclude.

2. The charge operator

To build the charge operator, we have to recognize the gravitational field as a fundamental field and see what implications we have on Qe and Qm. For example, in Ref. [12], the role of Qe

1 Recently, it has been discovered that spin ices, frustrated magnetic systems, have effective quasiparticle excitations with magnetic charges very close to magnetic monopoles [9].

2 For example, in SI units

4nh cf0 137

and Qm has been played by a cosmological constant interpreted as an eigenvalue of an associated Sturm-Liouville problem. To do this, we have introduced the Wheeler-DeWitt equation (WDW) [13] by rearranging the Einstein's field equations, to get:

Ha = (2K)Gijklnjnkl - 2g (3R - 2AC) = 0, (10)

for the sourceless case and in presence of a cosmological term.

Hq = (2K)Gijklnijnkl - ^ (3R - Hm) = 0, (11)

with a matter term and in absence of a cosmological constant. Note the formal similarity between Eqs. (10) and (11). Gijkl is the supermetric defined as

Gijkl = 2Qg(gikgjl + gilgjk - gijgkl) (12)

and 3R is the scalar curvature in three dimensions. n'j is called the supermomentum. This is the time-time component of the Einstein's field equations. It represents the invariance under time reparametrization and it works as a constraint at the classical level. Fixing our attention on the constraint (11), the explicit form of HM is easily obtained with the help of Eqs. (5) and (6), where one finds

U ~>„-T ,,a„P K Q2 + Qm

Hm = 2Kiapu u = ---4-.

Thus, the classical constraint Hq becomes

Hq = (2K)Gl]kln'jnkl - ^(3R - = 0. (14)

For a spherically symmetric metric described by (2) with g\(E/Ep) = g2(E/EP) = 1, it is easy to recognize that the classical constraint reduces to

3 r = 2G Q2 +4 <QQn b'(r) = GQ2 +2 <QQn , (15)

whose solution represents the Reissner-Nordstrom (RN) metric if

N2(r) =

b(r) = 2MG - G(Q2 + Q2m). (17)

On the other hand, changing the point of view, one could fix the background to see if there are other combinations solving the classical constraint (15). For example, if one fixes the background metric to be the Schwarzschild metric, one finds that the only solution compatible with the classical constraint is the trivial solution Qe = Qm = 0. The same situation happens for the de Sitter (dS) and Anti-de Sitter (AdS) metric. Things can change if we consider quantum fluctuations of the gravitational field. Indeed, these can be a source of nontrivial solutions as shown in Ref. [7]. To this purpose, we promote Hq to be an operator and the WDW equation in the presence of an electromagnetic field becomes

(2K)Gijkinijnk

3R K Q2 + Q

* = 0-

The WDW equation can be cast into the form

Q z* gj ] = -8npf (Q2 + Q2m )* gj ]•

Qz = (2K)Gijkinijnkl - 3R

is the charge operator. Now we see that this equation formally looks like an eigenvalue equation. To further proceed, we multiply Eq. (19) by * *[g ij ] and we functionally integrate over the three spatial metric gij, to obtain

D[gj]**[gu]( Vg^+r^ )*[gj]-

f D[gij ]* *[gij ]Q z* [gij ] = -8n f Finally one can formally re-write the WDW equation as

(* I fzd 3xQ z\*) _ Q2 + Q2m j* I Izd 3x(Vg/r4)\*)

{* I*) 8n (* I*) ,

where we have integrated over the hypersurface Z. The l.h.s. of Eq. (22) can be interpreted as an expectation value and the r.h.s. can be regarded as the associated eigenvalue with a weight. In principle, one should expand in perturbations even the determinant to one loop. This means that

(* I Izd'xQ Z I*) Q2 + Q2m {* \fz

d3xyg(0)+yg(1)+yg(2)+...+yg('') **)

(* I*)

(* I*)

where is the order of the approximation. However one may also adopt an alternative approach, where one fixes the background on the l.h.s. of Eq. (22) and consequently let the quantum fluctuations evolve, and then one verifies what kind of solutions one can extract from the r.h.s. in a recursive way. Therefore the first step begins with the r.h.s. of Eq. (22) which further can be reduced to

(* Ifzd3xQ z I*) Q2 + Q

(* I*)

If Eq. (24) gives the desired nontrivial eigenvalues at zero order, it means that an electric or magnetic charge (or both) has been created. This means that after the charge creation, the correct background will be represented by a Reissner-Nordstrom metric. This also means that Eq. (24) cannot be used anymore. In the next section we will discuss some subtleties arising in dealing with quantum fluctuations of the determinant of the r.h.s. of Eq. (22). If we consider Q^O^) as eigenvalues of the Sturm-Liouville problem for some fixed background, we unavoidably find that the one loop calculation is plagued by UV divergences. Therefore a regularization/renormal-ization scheme is needed to remove the divergences [7]. Nevertheless, the purpose of this paper is to propose a procedure to avoid such a scheme: the computation of 02(0^) in presence of Gravity's Rainbow which introduces only one scale, the Planck scale EP.

3. The charge operator in presence of Gravity's Rainbow

To compute the electric/magnetic charge in Gravity's Rainbow, we begin with the line element (2). The form of the background is such that the shift function

Ni = -Nui = g4 = 0 (25)

vanishes, while N is the previously defined lapse function. Thus the definition of Kij implies

= -j = gjEK (26)

j 2N g2(E) ij,

where the dot denotes differentiation with respect to the time t and the tilde indicates the quantity computed in absence of Rainbow's functions gi(E) and g2(E). For simplicity, we have set EP = 1 in g1(E/EP) and g2(E/EP) but later we will bring it back for relative comparison. The trace of the extrinsic curvature, therefore becomes

K = gijKij = g1(E)K (27)

and the momentum nij conjugate to the three-metric gij of £ is

nij = (Kgij - Kij) = (28)

2k g2(E)

Recalling that um = (-N, 0, 0, 0), in presence of Gravity's Rainbow we have the following modification

uM = uM = g1(EU (29)

which is useful to compute the distorted electromagnetic energy-momentum tensor. Indeed, from Eqs. (5) and (6), we find

g22(E)g11 (F )2u»,ri>2(F)- 1 Q2

l^(f01)u ug2(E) = 8n~ for the electric charge, while when we consider the magnetic charge, we get

W = (F23)2uMuvg22g33g1 (E)g4(E) = -L <tg\{E). (31)

8ng2(E) 8n r4

Since the scalar curvature R has the following property

R = gijRij = g2(E)R, (32)

we find that the WDW equation becomes

TtJ,vu>V = g2 (Jg (F01)2uVg2(E) = —Q g2(E)g2(E), (30)

(E)G nijnkl VI (R K Q2 ()g(E) 71 - 2gE)\R - ^ Q2g1;mg2

V = 0, (33)

where we have defined

Q2g1 ;mg2 = Q2eg2(E) + Q2mg2^(E) (34)

and where

1 G ijki

Gijti = -j—gigikgji + gilgjk - gijgki) = (35)

By repeating the same steps that have led to Eq. (22), we find

(№ I fsd3x Q x№) _ gfE № I fxd 3xQ2gl-,mg2 ^ ) {№\V) 8n g2(E) {№\V) '

where we have defined the distorted charge operator

Qs = KlE-Gmi nkl--rgg- R. (37)

g3,(E) J 2Kg2(E)

Since Eq. (36) as well as Eq. (22) cannot be solved exactly, we adopt a variational procedure with trial wave functionals of the Gaussian type. To further proceed, we fix a background metric gij and we consider quantum fluctuation around the background of the form gij = gij + hij. Following the procedure in Ref. [5], we canonically separate the degrees of freedom and since only the transverse traceless (TT) tensor contribution becomes relevant, we find

Q s = 4f Gj

3„ ijki -1±/ N , 1

g3(E) ( ' )jk 2Kg2(E) L ( ' )ijk[

where we have functionally integrated over Gaussian trial wave functionals. ALL represents the modified Lichnerowicz operator whose expression is

A= (ALh±)ij - 4RthiJ + 3Rhjj. (39)

Now when we consider the eigenvalue equation

(A Lh1-) t] = Eh (40)

we find

(Afh1).. = -—hi (4i)

1 L >lJ glE) lJ K ;

and the propagator K 1(x, y)lakl can be represented as

K 1(X, y)iakl = K1 (X, y)iakl = £ (42)

^ 2X(r)g4 (E)

where h('0>1(X) are the eigenfunctions of AL. t denotes a complete set of indices and X(t) are a set of variational parameters to be determined by the minimization of Eq. (38). The expectation value of QX is obtained by plugging the propagator in Eq. (38) and minimizing with respect to the variational function X(t). Therefore the one-loop charge in Gravity's Rainbow for the TT tensors is

Qs = -2 £gi(E)g2(E)L/E?(r) + ye2(z)

= _L iSELid^xJTg ^gipi. (44)

It is important to remark that if we had considered quantum fluctuations of the r.h.s. of Eq. (22), then the r.h.s. of Eq. (41) would have been modified with the introduction of the charge term which is possible only when Eq. (24) is solved. The expression in Eq. (43) makes sense when E2(r) > 0, where E2 are the eigenvalues of Am. Using the WKB approximation as used by't Hooft in the brick wall problem we can evaluate Eq. (43) explicitly. Extracting the energy density, we can write

•>2 2 oo 3

)2 1 2 p J r c2 To

1 gj(E) Qegi mg2 =--1 y i Eigl(E)g2(E^^\-E^ - m2(r)

2 g2(E) r4 3n2 j )g dEi i( )

l = 1E*

dEi, (45)

Lg2(E)

where E* is the value which annihilates the argument of the root and where we have defined two r-dependent effective masses m\(r) and m2(r)

2 6 ( b(r) \ 3 , 3

m2(r) = rH1 - b(r) + ^b(r) - b(r)

(r = r(x). (46)

2 b(r) 1 3

m2(r) = ^ - b^) + ^b'(r) + ^b(r)

We have hitherto used a generic form of the background. We now fix the attention on some backgrounds which have the following property

m2(r) = m2(r) = -m\(r), Wr e (rt,n). (47)

For example, the Schwarzschild background represented by the choice b(r) = rt = 2MG satisfies the property (47) in the range r e[rt, 5rt/4]. Similar backgrounds are the Schwarzschild-de Sitter and Schwarzschild-Anti-de Sitter. On the other hand, other backgrounds, like dS, AdS and Minkowski have the property

m2(r) = m2(r) = m\(r), Wr e (rt, o). (48)

3.1. The Schwarzschild case

Before going on, we examine the classical constraint for the Schwarzschild metric. From Eq. (33), the condition H = 0 reduces to

Q2 C)2 C)2

= 0 Qtg2(E) + Of g22(E) = 0, (49)

leading to the only trivial classical solution Qe = Qm = 0. Note that even in Minkowski space, we have a trivial solution with vanishing charges. This situation persists even at the quantum level, because there is no parameter which fixes the scale like the Schwarzschild mass can do. To further proceed, we observe that the Schwarzschild background satisfies condition (47) and Eq. (45) becomes

1 gf(E/Ep> 1 (50)

2 g2(E/Ep) r4 3n2

l+ = 3 E gi(E/Ep)

E2 2 d ( E

' —-+ m0(r)- -

g2(E/Ep) 0( )dE\g2(E/Ep)

l- = 3 E gi(E/Ep)

g22(E/Ep)

— m2(r)-1 ,

0( )dE\g2(E/Ep)J

For convenience we have reintroduced the Planck energy scale Ep in Eqs. (51) and (52). It is clear that the final result is strongly dependent on the choices we can do about g1(E/EP) and g2(E/EP). Nevertheless, some classes of functions cannot be considered because they do not lead to a finite result. For example, fixing

gi(E/Ep) = 1 — n(E/Ep)n and g2(E/Ep) = 1,

with n a dimensionless parameter and n an integer [8], Eq. (50) does not lead to a finite result and therefore will be discarded. Other examples that we have to discard without involving a specific form of g\(E/Ep) and g2(E/EP) are:

g2(E/Ep) = gî(E/Ep),

g2(E/Ep) = g—2(E/Ep)

g—2 (E/Ep) = gi(E/Ep).

When we adopt the choice (54), the electric charge becomes independent on the Rainbow's functions and Eq. (50) becomes

(ß2 + Qlg6(E/Ep)) = --— (l+ + I-).

2re 1V ' ^ 3n2

In order to have real results, the argument of the square root in I- must be positive for E ^ EP and this happens when g1(E/EP ) is of the form

gi(E/Ep) = + E/Ep

leading to a divergent result. The same situation happens for choice (55) where the magnetic charge becomes independent on the Rainbow's functions

^ {Q2eg6(E/Ep) + Ql) = (l+ + I-).

To have real results for E ^ Ep we have to impose

gi(E/Ep) = (1 + E/Ep)- 1

but also in this case I+ and I- diverge. Finally for the choice (56) we find that the integrals I+ and I- in Eq. (50) become finite but with a negative sign in front of the r.h.s. of Eq. (50) which means that Q2 (Qm) should be everywhere negative, a result which is not compatible with observation. Since the choices (54)-(56) do not give the desired result, we need to fix independently the form

of g1(E/EP) and g2(E/EP). It is immediate to observe that g1(E/EP) must have a shape such that I+ and I- be convergent. Following Ref. [5], we consider

g1(E/EP) =( 1 + ßE) expf-a^2

g2(E/Ep) = 1 a,ß e R.

In this configuration, we know that the integrals I+ and I- are finite and the introduction of the parameter j allows to change the sign in the one loop term. Note that the choice of the Gaussian was dictated by a comparison with a cosmological constant computation in the framework of Noncommutative theory [14]. By defining

and following the same steps of Ref. [5], one finds

QX(E/Ep) + Qjgf (E/Ep) 2Ep r4

= -2^ f(a,ß ; x),

f(a,ß; x) =

— cosh a

3x 2a2

sinh(ax2) +--— cosh (ax2)

eax erf (Vax)

and where K\(x) is the Bessel function of the first kind and erf (x) is the error function. For the Schwarzschild background, Eq. (62) becomes

lm0(r) EP

3MG r 3EP

8(MG)2Ep

r> 2MG r = 2MG

and its behavior is

i œ when M ^ 0 for r = 2MG,

0 when M ^ 0 for r> 2MG,

when M when M

œ for r = 2MG, œ for r> 2MG.

The behavior in Eq. (67) will be discarded, because it does not represent a physical realization. Therefore, we fix our attention on Eq. (66). For large x, the r.h.s. of Eq. (63) becomes:

g2 (E/Ep)Q^gi ;mg2 2Ep r4

(2ßa3/2 + Jna2)x 8ßa5/2 + 3^na3

4n 2a7/2 3 16ßa7/2

16n 2a11/2x

■ 5vna4

128n 2

a15/2x3

+ O(x -4)

while for small x we obtain

g\(E/Ep)Q2

eg1',mg2

4a5/2 + 3^/nßa2 (2jay + 2^0. ln^ aj~e)- 2jnß)x

2Ep r4

4n 2a9/2

16n 2^/a

15n 2'

+ O(x7),

where y is the Euler's constant. To keep the procedure as general as possible, in Eq. (63) we have kept the combination between the electric and magnetic charge coming from the energy-momentum tensor expressed in Eq. (34). It is also interesting to note that every choice we can do on the function g1(E/EP) satisfying the assumption (61), the magnetic monopole is less suppressed in the trans-Planckian region with respect to the electric charge. This could confirm that the magnetic monopole is a problem related to the very early universe. On the other hand, when we are on the cis-Planckian region, the electric charge and the magnetic monopole are not suppressed by the Rainbow's functions and behave in the same way. For this reason, we can first study the electric charge setting qL = 0. It is straightforward to see that if we fix

ß = —

then the linear divergent term of the asymptotic expansion (68) disappears and Eq. (63) vanishes for large x, namely when r = rt = 2MG and M ^ 0. Therefore on the throat rt one gets

Q2(a,ß,rt) =

By imposing that,

r4E4 rt Ep

f\a, —

2 '2r2E2

Q2(a, ß, n) =-= 0.73 x 10-2,

Qe( ^ t) 137 ,

then we find

h = 0.295/Ep,

where we have fixed a = 1/4. The situation is not much different if we choose

P = -3V

Indeed, Eq. (63) becomes

Q2(a,ß,rt) = -

r4E4 rt E p

f[a, -

3 v n 2r?Ep

and fixing again Q2 like in Eq. (72), one finds rt = 0.571/Ep.

Both the solution require a sub-Planckian throat. It is clear that the comparison of the fine structure constant 1/137 with Q2m is not possible. However later we will discuss how the charge operator Qs can give information about the magnetic monopole. If we move to the region where 5rt/4 > r > rt, we introduce a dependence on the radius r, which can be eliminated with the computation of

However when we choose the parametrization (70), the solution of Eq. (77) is imaginary and therefore will be discarded. On the other hand, when we choose parametrization (74), we find that the expression (69) reduces to

Q22 = -4n2

1 ■ «e^+11») «M*2 + O ((2MG)5/2)

and the computation of Eq. (77) in this case leads to the following relationship

2exm Y + 1

Since r e[rt, 5rt/4], this implies that

— expl y + —

— expl y + —

then we find the following bound

2.9638 x 10-2 =

33 64n 2

> Q2(a, r) >

= 3.9908 x 10"

3.2. The de Sitter (Anti-de Sitter) case

Even in this case, we examine the classical constraint for the dS and AdS metric, respectively. For the AdS metric it is immediate to verify that the condition H = 0 reduces to l2

R- c^m2 = 0 g(4g2(E) + ±mmg22(E)) = - AAdS,

Q« -2rv\ I Qm „2/

4nr4 \ r4 * r

which is never satisfied, while for the dS metric, we find

eg1 ;mg2

Qg2(E) + g%(E) ) = AdS.

4nr4 \r4 * r4

Moreover, if we fix the radius to the value r = V3/AdS, we find

GAds(Q2egl{E) + Q2mg22(E)) = 9, (84)

which fixes the values of Qe, Qm and AdS to values incompatible with observation. However, things can be different from the quantum point of view. Since in the dS and AdS cases, the condition (48) holds, Eq. (45) becomes

1 g2(E/Ep) Qle

eg1 ;mg2

2 g2(E/Ep) r4

where I- is given by Eq. (52). Choosing the Rainbow's functions like in the Schwarzschild case, one finds

g2(E/Ep)Q

eg1 ;mg2

2E4 r4

4a 2 n 2

3 + 2ax

2\e-ax2

- K1\—

where x is expressed by Eq. (62), but with a different m0(r). Indeed, we have

We can gain more information by evaluating the r.h.s. of Eq. (86) for small and large x. For large x, one gets

g? (E/EP)Q2gi;mg2 2Ep r 4

while for small x, we get

g1 (E/Ep)Qeg\;mg2

2Ep r4

2n 3/2a3/2

2n 3/2a3/2 4n 3/2a5/2

8n3/2a5/2 x 64n3/2a7/2 x3

+ O(x -5)

(4va+(2^/a+pjn) 2

+--. „ — x

4n 2a5/2

4n 2a3/2

^Valn(x4^va) + yVa - ^vn] 4 , 6,

--x + O(x .

It is interesting to note that the expression is finite for every x. Beginning with the dS case, we

0, x ■

observe that the range of the radius r is [0, V3/AdS ] and when r ■ because of behavior (88). On the other hand, when r

œ which is vanishing

V6/Ads, x ^ 0. However r = ^J6/Ads corresponds to a region external to the dS horizon which is unphysical and therefore will be discarded. Rather when

Therefore keeping the same parametrization that allows a vanishing contribution for small x, Eq. (86) becomes

Q2( a, -

3 V n EP !

3 + ~EA\) exp(. -

where we have excluded the trans-Planckian region which suppresses the charge contribution. By imposing that

4' 3Vn' Ep

we find that

A as ~ 16E2

and the corresponding "Cosmological radius" becomes

I 3 _ 0.43301

Concerning the AdS case, we observe that since r e [0, +m), when

^ x = —-. (95)

and Qe(1/4, -e/(3*Jn),^/AAdS/EP) ^ <x and therefore will be discarded. 4. Magnetic monopoles

As introduced in Section 2, our calculation applies also to magnetic monopoles. However, since we have no experimental evidence in high energy physics, we need to use the Dirac proposal between the magnetic monopole and the electric charge described by the relationship (7) to fix numbers. Therefore, it is immediate to see that

Qm = — = 73.543 Q2m = 5408.6. (96)

Since the value of Qm is quite large, for the Schwarzschild metric we can use parametrization (70) which keeps under control large values of x, while the parametrization (74) will be discarded. Setting

r4E% J1 JW 3

' Pf[ 4' "Y"; =5408.6' (97)

we find

= 7-. (98)

Note that when we compare rQm with rQe, we find

-Qm 6.6

= 22.373. (99)

rQe 0.295 On the other hand, if we use the dS metric, we find

QK 4,- ^ =5408.6, (100)

which implies

and the corresponding "Cosmological radius" becomes

3 11.18

rQm=i US=-ËT- (102)

Once again, when we compare rQm with rQe we find fQm 11.18

A- =-= 25.819. (103)

rQe 0.43301

5. Conclusions

that Gravity's Rainbow can be considered as a good tool for probing the spacetime foam picture suggested by Wheeler.

Appendix A. The electromagnetic energy-momentum tensor in SI units

The electromagnetic energy-momentum tensor in free space and in SI units is defined as 1

Tvv —

fvyfy 4 g/xvFygFY

where v0 is the vacuum permeability, Fvv — dvAv - dvAv is the electromagnetic field strength tensor. For a pure electric field, the electromagnetic potential Av assumes the form

A, — | Qe , 0,0,0) Fvv — F0i —--(A.2)

\4ne0rc J 4n^0rec

where €0 is the vacuum permittivity, c is the speed of light and Qe is the electric charge. On the other hand, for the pure magnetic field, the form is

Qm . r t^ Qn

cos 0) F,v — Fe3 —

where Qm is the magnetic charge measured in Ampere-meter (Am). Thus the T00 component of the energy-momentum tensor for the electromagnetic charges becomes

Av — 0, 0, 0, -|0 — cos0 F^v — Fe3 — l-l0~rm sin0, (A.3)

T00 — —f1 g11(F0i)e - 1 g00(geeg33)(Fe3)e

101 e e

1 ( 11 Q2

2|0 (4nr2)2

11 Qe 2 2

g - g00l0Qm

€0 c

1 \ .11Q2

g11 Q - g00l0Q2m\, (A.4)

Tivuluv — ———{ ^ + i0Q2m\. (A.5)

2(4nr2)2 ^ €0

where we have used the following relationship c2e0|0 — 1. With the help of the time-like vec tor uv, we obtain

ivuu — * (Q2 vv 2(4nr2)2 \ €0

For a spherically symmetric metric described by (2) with g1(E/EP) — g2(E/EP) — 1, it is easy to recognize that the classical constraint (11) reduces to

3 R — TaPuau? b'(r) — -^tA — + V0Qml, (A.6)

c4 4nr2c4 [ €0 J

whose solution represents the Reissner-Nordstrom metric if

N2(r) —

1_ Mxr-1

2MG G \ Q2 2 ,

^ — — - w{ ^ + V0Qlm\. (A.8)

Note that in CGS units, one defines €0 — (4n)-1 and v0 — 4n and the energy-momentum tensor is in agreement with the expression in (4).

References

[1] A.D. Sakharov, Dokl. Akad. Nauk SSSR, Ser. Fiz. 177 (1967) 70; A.D. Sakharov, Sov. Phys. Dokl. 12 (1968) 1040;

A.D. Sakharov, Sov. Phys. Usp. 34 (1991) 394; A.D. Sakharov, Gen. Relativ. Gravit. 32 (2000) 365.

[2] J.A. Wheeler, Phys. Rev. 97 (1955) 511.

[3] G. Amelino-Camelia, Int. J. Mod. Phys. D 11 (2002) 35, arXiv:gr-qc/0012051; G. Amelino-Camelia, Phys. Lett. B 510 (2001) 255, arXiv:hep-th/0012238.

[4] J. Magueijo, L. Smolin, Class. Quantum Gravity 21 (2004) 1725, arXiv:gr-qc/0305055.

[5] R. Garattini, G. Mandanici, Phys. Rev. D 83 (2011) 084021, arXiv:1102.3803 [gr-qc]; R. Garattini, G. Mandanici, Phys. Rev. D 85 (2012) 023507, arXiv:1109.6563 [gr-qc].

[6] R. Garattini, J. Cosmol. Astropart. Phys. 1306 (2013) 017, arXiv:1210.7760 [gr-qc].

[7] R. Garattini, Phys. Lett. B 666 (2008) 189, arXiv:0807.0082.

[8] Y. Ling, J. Cosmol. Astropart. Phys. 0708 (2007) 017, arXiv:gr-qc/0609129;

Y. Ling, X. Li, H. Zhang, Mod. Phys. Lett. A 22 (2007) 2749, arXiv:gr-qc/0512084; Y. Ling, B. Hu, X. Li, Phys. Rev. D 73 (2006) 087702, arXiv:gr-qc/0512083.

[9] C. Castelnovo, R. Moessner, S.L. Sondhi, Magnetic monopoles in spin ice, Nature 451 (2008) 42-45, http:// dx.doi.org/10.1038/nature06433;

L.D.C. Jaubert, P.C.W. Holdsworth, Signature of magnetic monopole and Dirac string dynamics in spin ice, Nat. Phys. 5 (2009) 258-261, http://dx.doi.org/10.1038/NPHYS1227;

S.T. Bramwell, S.R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, T. Fennell, Measurement of the charge and current of magnetic monopoles in spin ice, Nature 461 (2009) 956-959, http://dx.doi.org/10.1038/nature08500.

[10] P.A. Dirac, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 133 (1931) 60.

[11] H. Georgi, S. Glashow, Phys. Rev. Lett. 32 (1974) 438; G. 't Hooft, Nucl. Phys. B 79 (1974) 276;

A.M. Polyakov, JETP Lett. 20 (1974) 194.

[12] R. Garattini, Vestn. Tomsk. Gos. Pedagog. Univ. 44 (7) (2004) 72, arXiv:gr-qc/0409016; R. Garattini, J. Phys. Conf. Ser. 33 (215) (2006), arXiv:gr-qc/0510062.

[13] B.S. DeWitt, Phys. Rev. 160 (1967) 1113.

[14] R. Garattini, P. Nicolini, Phys. Rev. D 83 (2011) 064021, arXiv:1006.5418 [gr-qc].

[15] R. Garattini, Phys. Lett. B 685 (2010) 329, arXiv:0902.3927 [gr-qc].

[16] R. Garattini, F.S.N. Lobo, Phys. Rev. D 85 (2012) 024043, arXiv:1111.5729 [gr-qc].

[17] R. Garattini, M. Sakellariadou, Does Gravity's Rainbow induce inflation without an inflaton? arXiv:1212.4987 [gr-qc].

[18] R. Garattini, F.S.N. Lobo, Gravity's Rainbow induces topology change, arXiv:1303.5566 [gr-qc].